The correspondence theory of truth states that something is rendered true by the existence of a fact with corresponding elements and a similar structure. The theory maintains that the truth or falsity of a statement is determined only by how it relates to the world, and whether it accurately describes (i.e., corresponds with) that world. The theory presupposes an objective world and is therefore antagonistic to theories that problematise objectivity such as skepticism or relativism.

Problems with the theory arise from consideration of precisely what is supposed to correspond with what. If a statement is just a sentence then it is merely a physical thing (for example, ink on a page, or sound waves in the air) with no intrinsic meaning. Therefore it is usually claimed that it is the proposition (or meaning) expressed by a statement that is supposed to correspond with the facts. Yet both these "entities", propositions and facts, may be unappealing to minimalists who refuse to admit such abstract entities to their ontology. Also, precisely defining what constitutes correspondence is also a problem.

The proposition that P is true if and only if P corresponds with the facts.

So truth means correspondence with the facts. This is the traditional formulation of the theory. For example, it is true that some dogs bark if the proposition "Some dogs bark" corresponds with the fact that some dogs bark. For another example, the proposition that God exists is true if and only if the existence of God corresponds with the facts.

The most commonly cited problem for the correspondence theory is defining the relation of correspondence, and when a proposition corresponds with the facts. Bertrand Russell, and shortly after, Ludwig Wittgenstein, suggested that proposition and fact "correspond" when their structure is isomorphic. See Richard Kirkham's book cited below for a discussion of this view.

To get around this problem, we can easily see that in order for a proposition to be true according to the correspondence theory, there must exist some fact to which it corresponds: the proposition P has to correspond with the fact that P, if the proposition P is true. So, we can say that it is true that P if, and only if, there exists a fact that P. In this case, it is true that some dogs bark if, and only if, there exists a fact that some dogs bark.

The proposition that dogs bark is true if it is a fact that some dogs bark.

The proposition that God exists is true if it is a fact that God exists.

The proposition that snow is white is true if it is a fact that snow is white.

This solves the problem of defining correspondence by stating that if there is a fact that P, then that fact corresponds with the proposition that P. Basically, "true proposition" means "factual proposition."

However, this reformulation of the theory faces now a different problem: what are facts, and what does it mean to say that facts exist, or that there is some alleged fact? The theory now has to give some definition of what facts are.

There are at least two different ways to reply to this objection. The first way is to offer a theory of what facts are, which philosophers in the twentieth century have attempted to do. For example, facts are basically combinations of objects together with their properties or relations; so the fact that Fido barks is the combination of an object (i.e., Fido) with one of Fido's properties (that he barks).

The second way to reply is to note that the fact that Fido barks is only one type of fact. There are other types of facts, which may be facts about all dogs, or about the relation of dogs and cats. More importantly, it is possible to specify and categorize all those different kinds of facts. Therefore, a fact exists if all of its component parts exist. For example, if Fido exists, and Fido's barking exists, then the fact that Fido barks exists.